Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.192.3.77 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&10\\0&11\end{bmatrix}$, $\begin{bmatrix}7&4\\6&1\end{bmatrix}$, $\begin{bmatrix}7&4\\6&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2\times D_6$ |
| Contains $-I$: | no $\quad$ (see 12.96.3.i.2 for the level structure with $-I$) |
| Cyclic 12-isogeny field degree: | $2$ |
| Cyclic 12-torsion field degree: | $4$ |
| Full 12-torsion field degree: | $24$ |
Jacobian
| Conductor: | $2^{11}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 48.2.c.a, 72.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x y t + 2 x z t + x w t + 2 y^{2} t + y z t - y w t $ |
| $=$ | $3 x y t + x w t + y^{2} t - y z t$ | |
| $=$ | $x^{2} y + 2 x^{2} z + x^{2} w + x z w + x w^{2} + y^{3} + y^{2} z + y^{2} w + y z^{2} + y z w - y w^{2}$ | |
| $=$ | $x y w + 2 x z w + x w^{2} + 2 y^{2} w + y z w - y w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{5} + 4 x^{4} z + 6 x^{3} y^{2} + 5 x^{3} z^{2} + 12 x^{2} y^{2} z + 3 x^{2} z^{3} + \cdots - 3 y^{2} z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -3x^{7} - 15x^{6} - 21x^{5} - 30x^{4} - 21x^{3} - 15x^{2} - 3x $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(0:1/3:1/3:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2\cdot3\cdot5^4}\cdot\frac{1960827750000000000x^{2}t^{12}-20495021073705608478xw^{13}-236063668067690636784xw^{11}t^{2}+1402665701510147273112xw^{9}t^{4}-920071925209295390736xw^{7}t^{6}-2932564610613962340864xw^{5}t^{8}+2443047629638970303424xw^{3}t^{10}-2177280000000000000xwt^{12}-1341294728256yz^{13}-14299419214080yz^{11}t^{2}+1382632158640320yz^{9}t^{4}+6086314158926400yz^{7}t^{6}+1362393641637564480yz^{5}t^{8}+18177160014914100480yz^{3}t^{10}-477332168840104475520yzt^{12}-4251104132764767624yw^{13}+79863253005997478938yw^{11}t^{2}-58617684965627442656yw^{9}t^{4}-1287221182222446088224yw^{7}t^{6}+2347776341771482872864yw^{5}t^{8}+2599875843797339760672yw^{3}t^{10}-4224820329729580988160ywt^{12}-57815112096z^{14}-10961618274432z^{12}t^{2}+1585871981443008z^{10}t^{4}+22871127825866880z^{8}t^{6}-362512502524404288z^{6}t^{8}-16261237333083028608z^{4}t^{10}-22747893489736847331z^{2}w^{12}+110103201324503230152z^{2}w^{10}t^{2}+54348812994297874680z^{2}w^{8}t^{4}-821482124775971215392z^{2}w^{6}t^{6}+589667208033576147696z^{2}w^{4}t^{8}+893090816707543906176z^{2}w^{2}t^{10}-547290014988512654208z^{2}t^{12}-27419348142004094385zw^{13}+146487884297591144882zw^{11}t^{2}+160663952182939029524zw^{9}t^{4}-1959717927408265285968zw^{7}t^{6}+1720457754530942664144zw^{5}t^{8}+3214641363852091571424zw^{3}t^{10}-2966800918078013564736zwt^{12}+13084361243227340670w^{14}+975755009118791064w^{12}t^{2}+13634896916306545932w^{10}t^{4}-748363828093480303872w^{8}t^{6}+1158851354264883409248w^{6}t^{8}+1744501567233032445312w^{4}t^{10}-2282271249788240725440w^{2}t^{12}+425250000000000t^{14}}{t^{4}(122130763800xw^{9}-2442885708000xw^{7}t^{2}+12333279989808xw^{5}t^{4}-26050295322144xw^{3}t^{6}+527609376yz^{9}+2395552320yz^{7}t^{2}-21856353120yz^{5}t^{4}+72436744800yz^{3}t^{6}+9551871241920yzt^{8}-2922997779yw^{9}+26282081654yw^{7}t^{2}+1345758934572yw^{5}t^{4}-8292541057272yw^{3}t^{6}+35452542965760ywt^{8}+634160016z^{10}+8829741312z^{8}t^{2}+104357901312z^{6}t^{4}+1269035812800z^{4}t^{6}-26612737653z^{2}w^{8}+10522835496z^{2}w^{6}t^{2}+1586710294704z^{2}w^{4}t^{4}-11350101643488z^{2}w^{2}t^{6}+12212937405648z^{2}t^{8}-18329022444zw^{9}+20922928504zw^{7}t^{2}+3647580962976zw^{5}t^{4}-19138141768032zw^{3}t^{6}+37376210673216zwt^{8}+10274291250w^{10}-78587997714w^{8}t^{2}+566302038972w^{6}t^{4}-7241176875672w^{4}t^{6}+21857939141040w^{2}t^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.96.3.i.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{5}+6X^{3}Y^{2}+4X^{4}Z+12X^{2}Y^{2}Z+5X^{3}Z^{2}+3XY^{2}Z^{2}+3X^{2}Z^{3}-3Y^{2}Z^{3}+XZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 12.96.3.i.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x^{3}t+4x^{2}yt+xy^{2}t-y^{3}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x-y$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.0-12.a.2.12 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.96.1-12.d.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 12.96.1-12.d.1.5 | $12$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 12.96.2-12.a.1.2 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 12.96.2-12.a.1.3 | $12$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.384.5-12.d.2.2 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.384.5-12.d.2.3 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.384.5-12.e.1.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.384.5-12.e.4.4 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 12.576.13-12.m.1.7 | $12$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 24.384.5-24.bt.2.5 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.bt.4.2 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.384.5-24.ca.3.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.ca.4.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 36.576.13-36.i.1.6 | $36$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 36.576.19-36.n.1.6 | $36$ | $3$ | $3$ | $19$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 36.576.19-36.r.1.4 | $36$ | $3$ | $3$ | $19$ | $0$ | $1^{8}\cdot4^{2}$ |
| 60.384.5-60.x.1.7 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.384.5-60.x.3.3 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.384.5-60.y.1.4 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.384.5-60.y.2.5 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.960.35-60.bc.2.4 | $60$ | $5$ | $5$ | $35$ | $2$ | $1^{16}\cdot2^{4}\cdot8$ |
| 60.1152.37-60.cw.2.1 | $60$ | $6$ | $6$ | $37$ | $2$ | $1^{16}\cdot2\cdot4^{2}\cdot8$ |
| 60.1920.69-60.fk.1.6 | $60$ | $10$ | $10$ | $69$ | $6$ | $1^{32}\cdot2^{5}\cdot4^{2}\cdot8^{2}$ |
| 84.384.5-84.x.1.7 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.x.4.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.y.2.6 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.384.5-84.y.3.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.pz.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.pz.4.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.qg.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.qg.4.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.x.1.7 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.x.3.3 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.y.1.4 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.384.5-132.y.2.5 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.x.1.7 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.x.4.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.y.1.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.384.5-156.y.3.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.pz.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.pz.3.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.qg.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.qg.3.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.x.1.7 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.x.3.3 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.y.1.4 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.384.5-204.y.2.5 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.x.1.7 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.x.4.6 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.y.1.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.384.5-228.y.3.4 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.pz.1.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.pz.3.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.qg.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.qg.3.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.x.2.3 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.x.3.7 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.y.1.4 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.384.5-276.y.3.5 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.pz.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.pz.4.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.qg.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.qg.4.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |